1.3: Clinical Studies
Now you know how to collect a sample, next you need to learn how to conduct a study. We will discuss the basics of studies, both observational studies and experiments.
Observational Study: This is where data is collected from just observing what is happening. There is no treatment or activity being controlled in any way. Observational studies are commonly conducted using surveys, though you can also collect data by just watching what is happening such as observing the types of trees in a forest.
| Survey: Surveys are used for gathering data to create a sample. There are many different kinds of surveys, but overall, a survey is a method used to ask people questions when interested in the responses. Examples of surveys are Internet and T.V. surveys, customer satisfaction surveys at stores or restaurants, new product surveys, phone surveys, and mail surveys. The majority of surveys are some type of public opinion poll. |
Experiment: This is an activity where the researcher controls some aspect of the study and then records what happens. An example of this is giving a plant a new fertilizer, and then watching what happens to the plant. Another example is giving a cancer patient a new medication, and monitoring whether the medication stops the cancer from growing. There are many ways to do an experiment, but a clinical study is one of the more popular ways, so we will look at the aspects of this.
Clinical Study: This is a method of collecting data for a sample and then comparing that to data collected for another sample where one sample has been given some sort of treatment and the other sample has not been given that treatment (control). Note: There are occasions when you can have two treatments, and no control. In this case you are trying to determine which treatment is better.
Example \(\PageIndex{1}\): Clinical Study Examples
Here are examples of clinical studies.
- A researcher may want to study whether or not smoking increases a person's chances of heart disease.
- A researcher may want to study whether a new antidepressant drug will work better than an old antidepressant drug.
- A researcher may want to study whether taking folic acid before pregnancy will decrease the risk of birth defects.
Clinical Study Terminology :
Treatment Group: This is the group of individuals who are given some sort of treatment. The word treatment here does not necessarily mean medical treatment. The treatment is the cause, which may produce an effect that the researcher is interested in.
Control Group: This is the group of individuals who are not given the treatment.
Sometimes, they may be given some old treatment, or sometimes they will not be given anything at all. Other times, they may be given a placebo (see below).
Determine the treatment group, control group, treatment, and control for each clinical study in Example \(\PageIndex{1}\).
Solution
- A researcher may want to study whether or not smoking increases a person's chances of heart disease.
The treatment group is the people in the study who smoke and the treatment is smoking. The control group is the people in the study who do not smoke and the control is not smoking.
- A researcher may want to study whether a new antidepressant drug will work better than an old antidepressant drug.
The treatment group is the people in the study who take the new antidepressant drug and the treatment is taking the new antidepressant drug. The control group is the people in the study who take the old antidepressant drug and the control is taking the old antidepressant drug. Note: In this case the control group is given some treatment since you should not give a person with depression a non-treatment.
- A researcher may want to study whether taking folic acid before pregnancy will decrease the risk of birth defects.
The treatment group is the women who take folic acid before pregnancy and the treatment is taking folic acid. The control group is the women who do not take folic acid before pregnancy and the control is not taking the folic acid.
Note: In this case, you may choose to do an observational study of women who did or did not take folic acid during pregnancy so that you are not inducing women to avoid folic acid during pregnancy which could be harmful to their baby.
Confounding Variables: These are other possible causes that may produce the effect of interest rather than the treatment under study. Researchers minimize the effect of confounding variables by comparing the results from the treatment group versus the control group.
Controlled Study: Any clinical study where the researchers compare the results of a treatment group versus a control group.
Placebo: A placebo is sometimes used on the control group in a study to mimic the treatment that the treatment group is receiving. The idea is that if a placebo is used, then the people in the control group and in the treatment group will all think that they are receiving the treatment. However, the control group is merely receiving something that looks like the treatment, but should have no effect on the outcome. An example of a placebo could be a sugar pill if the treatment is a drug in pill form.
Example \(\PageIndex{3}\): Placebo Examples
For each situation in Example \(\PageIndex{1}\), identify if a placebo is necessary to use.
- A researcher may want to study whether or not smoking increases a person's chances of heart disease.
In this example, it is impossible to use a placebo. The treatment group is comprised of people who smoke and the control group is comprised of people who do not smoke. There is no way to get the control group to think that they are smoking as well as the treatment group.
- A researcher may want to study whether a new antidepressant drug will work better than an old antidepressant drug.
In this example, a placebo is not needed since we are comparing the results of two different antidepressant drugs.
- A researcher may want to study whether taking folic acid before pregnancy will decrease the risk of birth defects.
In this example, the control group could be given a sugar pill instead of folic acid. However, they may think that they are taking folic acid and so the psychological effect on a person's health can be being measured. This way, when we compare the results of taking folic acid versus taking a sugar pill, we can see if there were any dramatic differences in the results.
Blind Study: Usually, when a placebo is used in a study, the people in the study will not know if they received the treatment or the placebo until the study is completed. In other words, the people in the study do not know if they are in the treatment group or in the control group. This type of study is called a blind study. Note: When researchers use a placebo in a blind study, the people in the study are told ahead of time that they may be getting the actual treatment, or they may be getting the placebo.
Double-Blind Study: Sometimes when researchers are conducting a very extensive study using many healthcare workers, the researchers will not tell the people in the study or the healthcare workers which patients will receive the treatment and which patients will receive the placebo. In other words, the healthcare workers who are administering the treatment or placebo to the people in the study do not know which people are in the treatment group and which people are in the control group. This type of study is called a double-blind study.
Randomized Controlled Study: Any clinical study in which the treatment group and the control group are selected randomly from the population.
Whether you are doing an observational study or an experiment, you need to figure out what to do with the data. You will have many data values that you collected, and it sometimes helps to calculate numbers from these data values. Whether you are talking about the population or the sample, determines what we call these numbers.
Parameter: A numerical value calculated from a population
Statistic: A numerical value calculated from a sample, and used to estimate the parameter
Some examples of parameters that can be estimated from statistics are the percentage of people who strongly agree to a question and mean net worth of all Americans. The statistic would be the percentage of people asked who strongly agree to a question, and the mean net worth of a certain number of Americans.
Notation for Parameter and Statistics:
Parameters are usually denoted with Greek letters. This is not to make you learn a new alphabet. It is because there just are not enough letters in our alphabet. Also, if you see a letter you do not know, then you know that the letter represents a parameter. Examples of letters that are used are (mu), (sigma), (rho), and p (yes this is our letter because there is not a good choice in the Greek alphabet).
Statistics are usually denoted with our alphabet, and in some cases we try to use a letter that would be equivalent to the Greek letter. Examples of letter that are used are (x-bar), s, r, and (p-hat, since we already used p for the parameter).
In addition, N is used to denote the size of the population and n is used to denote the size of the sample.
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Sampling Error: This is the difference between a parameter and a statistic. There will always be some error between the two since a statistic is an estimate of a parameter. Sampling error is attributed to chance error and sample bias. Chance Error: The error inherent in taking information from a sample instead of from the whole population. This comes from the fact that two different samples from the same population will likely give two different statistics. |
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Sample Bias: The error from using a sample that does not represent the population. To avoid this, use some sort of random sample. Sampling Rate: The fraction of the total population that is in the sample. This can be denoted by n/N. |


4.1: Introduction
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It has long been known that pure carbon occurs in different forms (allotropes) including graphite and diamonds. But it was not until 1985 that a new form of carbon was recognized: buckminsterfullerene. This molecule was named after the architect and inventor R. Buckminster Fuller (1895–1983), whose signature architectural design was the geodesic dome, characterized by a lattice shell structure supporting a spherical surface. Experimental evidence revealed the formula, C 60 , and then scientists determined how 60 carbon atoms could form one symmetric, stable molecule. They were guided by bonding theory—the topic of this chapter—which explains how individual atoms connect to form more complex structures.


11.7: Test of Independence (2 of 3)
Learning Objectives
- Conduct a chi-square test of independence. Interpret the conclusion in context.
Here we continue our chi-square test of independence for the variables gender and body image in the population of U.S. college students.
Example
Gender and Body Image Continued
Step 1: State the hypotheses.
Here are our hypotheses from the previous page:
- H 0 : There is no relationship between gender and body image for U.S. college students. (The variables are independent.)
- H a : There is a relationship between gender and body image for U.S. college students. (The variables are dependent.)
Step 2: Collect and analyze the data.
If the variables are independent, the percentage of males and females with a given response will be the same or at least close. Previously, we determined that in our sample, there are differences in the percentage of males and females who answer “about right,” “overweight,” or “underweight.”
We need to determine if these differences are typical in random samples from a population where gender and body image are independent. Perhaps the differences we see in this sample are just fluctuations expected in random sampling. Or perhaps these differences are too large to be explained by chance. We will not know until we complete the hypothesis test.
Step 3: Assess the evidence.
We need to determine the expected values and the chi-square test statistic so that we can find the P-value.
Calculating Expected Values for a Test of Independence
Expected counts always describe what we expect to see in a sample if the null hypothesis is true. In this situation, if gender and body image are independent, then we expect the probability that a student answers “about right” in the sample to be the same probability that a male (or a female) student answers “about right” (similarly for “overweight” or “underweight” responses).
Here are the calculations of expected counts for the response “about right”:
- Probability that a student will answer “about right”: P (about right) = (855/1,200) = 0.7125
- Expected count of females in the sample who will answer “about right”: 0.7125(760) = 541.5
- Expected count of males in the sample who will answer “about right”: 0.7125(440) = 313.5
Here are the calculations of expected counts for the response “overweight”:
- Probability that a student will answer “overweight”: P (overweight) = (235/1,200) = 0.1958
- Expected count of females in the sample who will answer “overweight”: 0.1958(760) = 148.8
- Expected count of males in the sample who will answer “overweight”: 0.1958(440) = 86.2
Example
More on Gender and Body Image
Checking Conditions
The conditions for use of the chi-square distribution are the same as we learned previously:
- The sample is random.
- All of the expected counts are 5 or greater.
Since the data meets the conditions, we can proceed with calculating χ 2 test statistic.
Calculating the Chi-Square Test Statistic
We calculate the chi-square test statistic as we did in “Chi-Square Test for One-Way Tables.”
We will use technology to calculate the chi-square value. But for this sample, we will show the calculation.
Finding Degrees of Freedom and the P-Value
For a chi-square tests based on two-way tables, the degrees of freedom are
- (number of explanatory categories – 1) × (number of response categories – 1)
You will also see this written: ( r − 1)( c − 1), where r is the number of rows and c is the number of columns in the two-way table (when we write the table without row and column totals). In this case the degrees of freedom are (2 − 1)(3 −1 ) = 2.
We use the chi-square distribution with df = 2 to find the P-value. Note that the chi-square test statistic for this sample is so large that it is off the scale used in the simulation. So we conclude that the P-value is essentially zero.
Step 4: Conclusion
The relationship between gender and body image is statistically significant in this sample. We reject the null hypothesis and accept the alternative hypothesis. Gender and body image are dependent variables in the population of U.S. college students. (P-value is essentially 0.)
Try It
More on Gender and Body Piercing
A study was done on the relationship between gender and ear piercing among high-school students. A sample of 1,000 students was chosen, then classified according to both gender and whether or not they had either of their ears pierced. The following information is available:
https://assessments.lumenlearning.co...sessments/3727
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


 Before you watch, discuss the following questions in pairs or groups of three.
- These are pictures of Andy. He is a farmer. What do you think he does every day?
- Andy also cares a lot about the environment. Because of this, he practices sustainable agriculture. What is sustainable agriculture? If you don't know, you can guess.
- In the video, Andy will talk about his values (the things that are most important to him in life). What are your values?
Vocabulary
Vocabulary Task 1
Do you know these words? Discuss them with your partner. Then share what you know with the class. Use the pictures below to help you understand the ideas.
- sustainable
- sustainable agriculture
- industrial agriculture
- values
Notes:
- Windmills and solar panels are an example of sustainable energy sources because wind and sun cannot be used up completely or destroyed.
- Industrial agriculture is a type of farming that involves growing large numbers of plants and animals to be sold for food.
- Your values are the things that are most important to you. They can include family, money, education, music, art, etc. What is most important to you in life?
Vocabulary Task 2
Use these words to complete the table below:
| plant crops | pull weeds | watering | harvesting | pickup truck | fence | produce |
| add carbon to the atmosphere | challenging | vegan | meat production | pollute waterways | destroy | habitat |
| Number | Picture or definition | Word or phrase |
|---|---|---|
| 1 |
Photo by Brittany Zemlick |
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| 2 |
Photo by Nikoline Arns on Unsplash |
plant crops |
| 3 |
Photo by Markus Spiske on Unsplash |
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| 4 |
Photo by duong chung on Unsplash |
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| 5 |
raising animals for food Photo by Alison Marras on Unsplash |
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| 6 |
remove unwanted plants Photo by Nikoline Arns on Unsplash |
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| 7 |
a radio or music program your can download to your phone or computer from the internet |
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| 8 |
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| 9 |
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| 10 |
something that is difficult but may be interesting or enjoyable Photo by Jukan Tateisi on Unsplash |
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| 11 |
a place where animals live; for example, these frogs live in the water Photo by Ken Goulding on Unsplash |
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| 12 |
to cause something to end or no longer exist Photo by Ken Goulding on Unsplash |
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| 13 |
Photo by Alexandr Podvalny on Unsplash |
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| 14 |
a person who does not eat food that comes from animals (milk, meat, eggs) OR food that does not contain animal products Photo by Edgar Castrejon on Unsplash |
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| 15 |
Photo by Marcin Jozwiak on Unsplash |
While You Watch
Watch the video about Andy
Listen for the Main Idea
While you watch, answer the questions below. Takes notes. Then share your answers with your partner.
- What does Andy do every day in his job?
- What are his values?
Listen for Detail
Watch the video again and answer the multiple choice and fill in the gap questions.
- What's the man's name? _____________________
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Where did he grow up?
- Oxford
- San Diego
- El Cajon
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Where does he live now?
- Oxford
- San Diego
- Scotland
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How many brothers and sisters does he have?
- 1
- 2
- 3
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Where are his parents from?
- England and Poland
- El Cajon and San Diego
- Scotland and Poland
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What's his job?
- He's a farmer.
- He's a teacher.
- He's a vegan.
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What does he do every day? Choose all the correct answers.
- He waters plants.
- He pulls weeds.
- He plants crops.
- He repairs fences.
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Why does he like his job? Choose all the correct answers.
- He likes being outdoors in a beautiful place.
- He likes nature.
- He likes working to improve the environment.
- He likes office work.
- He likes working with the public.
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What doesn't he like about his job? Choose all the correct answers.
- He doesn't like being outdoors in a beautiful place.
- He doesn't like nature.
- He doesn't like working to improve the environment.
- He doesn't like office work.
- He doesn't like working with the public.
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Which one of these is NOT his hobby?
- He cooks.
- He runs.
- He plays the guitar.
- He listens to pod cats.
- He rides his bike.
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Does he have any pets?
- Yes, he does.
- No, he doesn't.
- He lives in a house with two cats.
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What kind of car does he drive?
- He drives a pick up truck.
- He drives a Toyota Camry.
- He drives a Jeep.
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Does he like waking up early in the morning?
- Yes, he does.
- No, he doesn't.
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Why doesn't he eat meat?
- He doesn't like the taste of meat.
- He feels eating meat hurts the environment.
- He eats meat every day.
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What are the most important things in his life? Choose all the correct answers.
- His family.
- Making the world a better place.
- Playing music.
- Running.
Interactive: Check your Answers
Select the correct answer, click "check" to see if your answer is correct, and then click the arrow to move to the next question. When you have answered all of the questions, click "Finished."
After You Watch
Post-Listening Discussion
Discuss and answer these questions with a partner. Be ready to explain why you chose each answer.
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What is the genre of this video?
- a TikTok video
- a movie
- a presentation
- an interview
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Who is the most likely audience? (Who will watch this video?)
- young children in school
- young people who like to travel
- teachers
- ESL students
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What is the purpose of this video?
- to teach children about farmer
- to persuade people to become vegans
- to model an interview for English Language students
- Would you like to have a job like Andy's? Why or why not?
- Tell your partner something interesting you learned from this video.
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If you interviewed a friend or family member about their job, what questions would you ask?
- Write a list of questions here.
- Then share your questions with your partner. Add some good questions from your partner's list.


1.1.7: Summary
A set is a collection of objects, the elements of the set. We write \(x \in A\) if \(x\) is an element of \(A\) . Sets are extensional —they are completely determined by their elements. Sets are specified by listing the elements explicitly or by giving a property the elements share ( abstraction ). Extensionality means that the order or way of listing or specifying the elements of a set doesn’t matter. To prove that \(A\) and \(B\) are the same set ( \(A = B\) ) one has to prove that every element of \(X\) is an element of \(Y\) and vice versa.
Important sets include the natural ( \(\Nat\) ), integer ( \(\Int\) ), rational ( \(\Rat\) ), and real ( \(\Real\) ) numbers, but also strings ( \(X^*\) ) and infinite sequences ( \(X^\omega\) ) of objects. \(A\) is a subset of \(B\) , \(A \subseteq B\) , if every element of \(A\) is also one of \(B\) . The collection of all subsets of a set \(B\) is itself a set, the power set \(\Pow{B}\) of \(B\) . We can form the union \(A \cup B\) and intersection \(A \cap B\) of sets. An ordered pair \(\tuple{x, y}\) consists of two objects \(x\) and \(y\) , but in that specific order. The pairs \(\tuple{x, y}\) and \(\tuple{y, x}\) are different pairs (unless \(x = y\) ). The set of all pairs \(\tuple{x, y}\) where \(x \in A\) and \(y \in B\) is called the Cartesian product \(A \times B\) of \(A\) and \(B\) . We write \(A^2\) for \(A \times A\) ; so for instance \(\Nat^2\) is the set of pairs of natural numbers.


3: Integers
At over 29,000 feet, Mount Everest stands as the tallest peak on land. Located along the border of Nepal and China, Mount Everest is also known for its extreme climate. Near the summit, temperatures never rise above freezing. Every year, climbers from around the world brave the extreme conditions in an effort to scale the tremendous height. Only some are successful. Describing the drastic change in elevation the climbers experience and the change in temperatures requires using numbers that extend both above and below zero. In this chapter, we will describe these kinds of numbers and operations using them.
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- 3.1: Introduction to Integers (Part 1)
- The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero. Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. So, in opposite notation, -a means the opposite of the number a. Integers are counting numbers, their opposites, and zero. The absolute value of a number is its distance from 0 on the number line.
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- 3.2: Introduction to Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
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- 3.3: Add Integers (Part 1)
- In order to understand how addition and subtraction of negative numbers works, we will use two color counters. We let a blue counter represent a positive number and a red counter will represent a negative number. When the numbers' signs are the same, the counters would be all the same color, so add them together. When the numbers' signs are different, some counters would make neutral pairs, so subtract them to see how many are left.
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- 3.4: Add Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
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- 3.5: Subtract Integers (Part 1)
- We continue to use the blue counters to represent positive numbers and the red counters to represent negative numbers. When there are enough counters of a certain color to take away, subtract. But, when there are not enough of the counters to take away, add neutral pairs and then subtract. The Subtraction Property states that that subtraction of signed numbers can be done by adding the opposite.
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- 3.6: Subtract Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
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- 3.7: Multiply and Divide Integers (Part 1)
- Multiplying or dividing two numbers with the same sign, positive or negative, will produce a positive number. While multiplying or dividing two numbers with different signs will produce a negative number. Also, multiplying or dividing a number by -1 gives its opposite.
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- 3.8: Multiply and Divide Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
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- 3.9: Solve Equations Using Integers; The Division Property of Equality (Part 1)
- To determine if a number is a solution to an equation, first substitute the number for the variable in the equation. Then simplify the expressions on both sides of the equation and determine whether the resulting equation is true or not. If it is true, the number is a solution. But if it is not true, the number is not a solution. The subtraction, addition, and division properties of equality can be used to find solutions to equations.
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- 3.10: Solve Equations Using Integers; The Division Property of Equality (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
Figure 3.1 - The peak of Mount Everest. (credit: Gunther Hagleitner, Flickr)


10.4: Method of inversion
Here is an application of inversion, which we include as an illustration; we will not use it further in the book.
Let \(ABCD\) be an inscribed quadrangle. Assume that the points \(A,B,C\), and \(D\) appear on the circline in the same order. Then
\(AB \cdot CD + BC \cdot DA = AC \cdot BD.\)
- Proof
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Assume the points \(A,B,C,D\) lie on one line in this order.
Set \(x = AB\), \(y = BC\), \(z = CD\). Note that
\(x \cdot z + y \cdot (x + y + z) = (x + y) \cdot (y + z).\)
Since \(AC = x + y\), \(BD = y + z\), and \(DA = x + y + z\), it proves the identity.
It remains to consider the case when the quadrangle \(ABCD\) is inscribed in a circle, say \(\Gamma\).
The identity can be rewritten as
\(\dfrac{AB \cdot DC}{BD \cdot CA} + \dfrac{BC \cdot AD}{CA \cdot DB} = 1.\)
On the left hand side we have two cross-ratios. According to Theorem 10.3.1a , the left hand side does not change if we apply an inversion to each point.
Consider an inversion in a circle centered at a point \(O\) that lies on \(\Gamma\) between \(A\) and \(D\). By Theorem Theorem 10.3.2 , this inversion maps \(\Gamma\) to a line. This reduces the problem to the case when \(A, B, C\), and \(D\) lie on one line, which was already considered.
In the proof above, we rewrite Ptolemy’s identity in a form that is invariant with respect to inversion and then apply an inversion which makes the statement evident. The solution of the following exercise is based on the same idea; one has to make a right choice of inversion.
Assume that four circles are mutually tangent to each other. Show that four (among six) of their points of tangency lie on one circline.
- Hint
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Label the points of tangency by \(X, Y , A, B, P\), and \(Q\) as on the diagram. Apply an inversion with the center at \(P\). Observe that the two circles that tangent at \(P\) become parallel lines and the remaining two circles are tangent to each other and these two parallel lines.
Note that the points of tangency \(A'\), \(B'\), \(X'\), and \(Y'\) with the parallel lines are vertexes of a square; in particular they lie on one circle. These points are images of \(A, B, X\), and \(Y\) under the inversion. By Theorem Theorem 10.3.1 , the points \(A, B, X\), and \(Y\) also lie on one circline.
Assume that three circles tangent to each other and to two parallel lines as shown on the picture.
Show that the line passing thru \(A\) and \(B\) is also tangent to two circles at \(A\).
- Hint
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Apply the inversion in a circle with center \(A\). The point \(A\) will go to infinity, the two circles tangent at \(A\) will become parallel lines and the two parallel lines will become circles tangent at \(A\); see the diagram.
It remains to show that the dashed line (\(AB'\)) is parallel to the other two lines.


Chapter 16: Unbundling Precarious Creativity in China – “Knowing-How” and “Knowing-To”
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Job security is under unprecedented threat in many developed nations as a consequence of the mechanization of work.¹ In addition, rising production costs are seeing the relocation of production to low-cost locations. This is a well-known story. Emerging economies are achieving substantial growth by providing cheap labor and preferential investment policies. For China, already an economic powerhouse, a foreign country’s insecurity is a their security: the “made in China” phenomenon manifests in products that are designed elsewhere and fabricated in China. Much of this outsourced production involves components. Economists call this “tradein-tasks,” “unbundling,”² or OEM (original equipment manufacturing).
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16.1: Introduction
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Introduction to the goals of the chapter: unbundling the concept of creative precarity and analyzing the relationship between creativity and knowledge capital, focusing on China.
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16.2: Contextualizing Precarious Creativity
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Contextualizing creative precarity in China, with an introduction to the concepts of “knowledge-how” and “knowledge-to.”
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16.3: A Chinese Political Economy Framework
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Considering the social and political context of labor in China, and how this distinguishes the meaning of precarious creativity in China from that in other countries.
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16.4: Knowledge Capital
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Definition of knowledge capital, and how it relates to both knowing-that and knowing-how.
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16.5: Media – The Game Changes
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Recent changes in the Chinese media industry, including a rise in workers trained in multinational companies transferring their knowledge to Chinese companies, the commercialization of broadcasting industries, and the growth of private-owned production companies which are still subject to state censorship.
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16.6: The Cultural Innovation Timeline Reconsidered
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An argument for the concept of creative precarity in China being not simply a negative indicator, using the cultural innovation timeline (standardized production, imitation, collaboration, cultural trade, production/management consolidation, peer/creative communities).
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16.7: Concluding Remarks – The (Precarious) Elephant in the Room
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How creative precarity in China is distinguished from creative precarity in other developed nations, by censorship and the existence of an additional dimension of knowledge capital (“knowing-to”).
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16.8: Notes
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4.1.1: Lots of Flags
Lesson
Let's explore the U.S. flag.
Exercise \(\PageIndex{1}\): Scaled or Not?
1. Which of the geometric objects are scaled versions of each other?
2. Pick two of the objects that are scaled copies and find the scale factor.
Exercise \(\PageIndex{2}\): Flags are Many Sizes
One standard size for the United States flag is 19 feet by 10 feet. On a flag of this size, the union (the blue rectangle in the top-left corner) is \(7\frac{5}{8}\) feet by \(5\frac{3}{8}\) feet.
There are many places that display flags of different sizes.
- Many classrooms display a U.S. flag.
- Flags are often displayed on stamps.
- There was a flag on the space shuttle.
- Astronauts on the Apollo missions had a flag on a shoulder patch.
- Choose one of the four options and decide on a size that would be appropriate for this flag. Find the size of the union.
- Share your answer with another group that used a different option. What do your dimensions have in common?
Exercise \(\PageIndex{3}\): What Percentage is the Union?
On a U.S. flag that is 19 feet by 10 feet, the union is \(7\frac{5}{8}\) feet by \(5\frac{3}{8}\) feet. For each question, first estimate the answer and then compute the actual percentage.
- What percentage of the flag is taken up by the union?
- What percentage of the flag is red? Be prepared to share your reasoning.
Are you ready for more?
The largest U.S. flag in the world is 225 feet by 505 feet.
- Is the ratio of the length to the width equivalent to \(1:1.9\), the ratio for official government flags?
- If a square yard of the flag weighs about 3.8 ounces, how much does the entire flag weigh in pounds?
Summary
Imagine you have a painting that is 15 feet wide and 5 feet high. To sketch a scaled copy of the painting, the ratio of the width and height of a scaled copy must be equivalent to \(15:5\). What is the height of a scaled copy that is 2 feet across?
| width | height |
|---|---|
| \(15\) | \(5\) |
| \(2\) | \(h\) |
We know that the height is \(\frac{1}{3}\) the width, so \(h=\frac{1}{3}\cdot 2\) or \(\frac{2}{3}\).
Sometimes ratios include fractions and decimals. We will be working with these kinds of ratios in the next few lessons.
Glossary Entries
Definition: Percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
Practice
Exercise \(\PageIndex{4}\)
A rectangle has a height to width ratio of \(3:4.5\). Give two examples of dimensions for rectangles that could be scaled versions of this rectangle.
Exercise \(\PageIndex{5}\)
One rectangle measures 2 units by 7 units. A second rectangle measures 11 units by 37 units. Are these two figures scaled versions of each other? If so, find the scale factor. If not, briefly explain why.
Exercise \(\PageIndex{6}\)
Ants have 6 legs. Elena and Andre write equations showing the proportional relationship between the number of ants, \(a\), to the number of ant legs \(l\). Elena writes \(a=6\cdot l\) and Andre writes \(l=\frac{1}{6}\cdot a\). Do you agree with either of the equations? Explain your reasoning.
(From Unit 2.2.2)
Exercise \(\PageIndex{7}\)
On the grid, draw a scaled copy of quadrilateral ABCD with a scale factor \(\frac{2}{3}\).
(From Unit 1.1.4)
Exercise \(\PageIndex{8}\)
Solve each equation mentally.
- \(\frac{5}{2}\cdot x=1\)
- \(x\cdot\frac{7}{3}=1\)
- \(1\div\frac{11}{2}=x\)
(From Unit 1.1.5)
Exercise \(\PageIndex{9}\)
Lin has a scale model of a modern train. The model is created at a scale of 1 to 48.
- The height of the model train is 102 millimeters. What is the actual height of the train in meters? Explain your reasoning.
- On the scale model, the distance between the wheels on the left and the wheels on the right is \(1\frac{1}{4}\) inches. The state of Wyoming has old railroad tracks that are 4.5 feet apart. Can the modern train travel on those tracks? Explain your reasoning.
(From Unit 1.2.5)


<IP_ADDRESS>: Il "si" impersonale e passivante
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Impersonal and passive si
Remember those useful phrases you learned in your first Italian class – Come si dice…? [ How do you say…? ] Come si scrive…? [ How do you write/spell…? ] and Come si legge…? [ How do you read/pronounce…? ] The word si has a general meaning of people, everybody, one, and an unspecified you , and it is often used in Italian to express actions that are common to many people.
|
1
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si + verb in the 3rd-person singular when the direct object that follows the action is singular | Al bar in Piazza Dante si guarda la partita di Coppa Campioni. |
|
2
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si + verb in the 3rd-person plural when the direct object that follows the action is plural | Al bar in Piazza Dante si guardano le partite di Coppa Campioni. |
|
3
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si + verb in the 3rd person singular with verbs that are not followed by a direct object | La domenica si va allo stadio. |
| 4 | ci si + verb in the 3rd-person singular with reflexive verbs | Ci si diverte allo stadio. |
There are theoretical differences between the passive si (in the first two cases above) and impersonal si (in the third and fourth cases above). Do not worry about them, as si means the same thing in both cases.
Notice that:
A. Si can also mean we , as in Si va? [ Shall we go? ].
B. With si , we use adjectives in the masculine plural form , because si has a plural connotation (people, everybody, one, an unspecified you), as in Quando si perde una partita si è delusi [ When one loses a game one is disappointed = When people lose a game people are disappointed ].
C. Si can be used with all tenses:
- Present tense: Oggigiorno si giocano alcune partite il sabato.
- Imperfetto: Un tempo si giocavano tutte le partite la domenica.
- Passato prossimo: Ieri sera si è giocato a calcio. In a compound tense one always uses essere with si .
- and other tenses you will learn.
D. Do not confuse the passive and impersonal si [ In questa lavanderia si lavano i cappotti ] with the reflexive si [ Lui si lava le mani] .


2 .
Discuss the maternal physiologic changes associated with the cardiovascular system and their causes.
3 .
Explain the role of progesterone in maintaining the pregnancy.
4 .
Differentiate between the presumptive, probable, and positive signs of pregnancy.
5 .
What is the long-term result when the actions by the pregnant person’s partner demonstrate support of the pregnancy?
6 .
What should the parents consider when observing the siblings’ reaction to the pregnancy?
7 .
Explain why the discomforts of pregnancy are common.
8 .
Describe three self-care activities to decrease lower back pain. Provide rationale for why you chose these activities.
9 .
What information would you provide to a pregnant patient about decreasing insomnia?
10 .
What is the purpose of surfactant?
11 .
Why is the embryo so vulnerable to the effects of teratogens?
12 .
Describe the steps of pre-embryonic development from conception to implantation.
13 .
Differentiate between a certified professional midwife and a traditional birth attendant.
14 .
What should the nurse do if a pregnant person asks for advice about how to choose a perinatal health care provider?
15 .
Differentiate between the health care provided at inpatient and free-standing birthing centers.
16 .
What should the nurse do if a pregnant person asks for advice about having a home delivery?


6.5: Notes
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- Violaine Roussel
- University of California Press
1 Robert R. Faulkner, Music on Demand: Composers and Careers in the Hollywood Film Industry (New Brunswick, NJ: Transaction, 1983).
2 On the effects of the division between core and non-core agencies in terms of market segmentation and artistic careers, see William T. Bielby and Denise D. Bielby, “Organizational Mediation of Project-Based Labor Markets: Talent Agencies and the Careers of Screenwriters,” American Sociological Review 64.1 (1999): 64–85.
3 My work is based on 112 interviews conducted in Los Angeles, mostly with agents and to a lesser extent with managers, production professionals, artists, lawyers, and publicists, as well as in situ observations at talent agencies or alongside an agent. This research received support from the European Commission (7th Framework Program, Marie Curie Fellowship).
4 Ron Meyer, William Haber, Michael Rosenfeld, and Rowland Perkins are the other four.
5 Below-the-line encompasses the technical professions working on film and television projects. See Laura Grindstaff and Vicki Mayer, “The Importance of Being Ordinary: Brokering Talent in the New-TV Era,” in Brokerage and Production in the American and French Entertainment Industries: Invisible Hands in Cultural Markets , ed. Violaine Roussel and Denise Bielby (Lanham, MD: Lexington Books, 2015), 131–152.
6 In addition to formalized divisions, various other principles of fragmentation combine. Agents are categorized much like their clients are, according to homologous (prestige) hierarchies.
7 Positions of “crossover agent” are created in some agencies for that purpose. The question of how to institutionalize the activity of social and digital media agents, in a context where no one initially knows how to monetize digital content, also illustrates the strength of the institutional framework of the big agencies. On the way film and television professionals approach this “digital revolution,” see Michael Curtin, Jennifer Holt, and Kevin Sanson, eds., Distribution Revolution: Conversations about the Digital Future of Film and Television (Berkeley: University of California Press, 2014).
8 In 2010, CAA formed a strategic partnership with the global private equity firm TPG Capital, which owns 35 percent of the agency. WME partnered with Silver Lake in May 2012, the private equity firm acquiring a 31 percent share in the agency.
9 The Wrap , April 13, 2014, www.thewrap.com/leaked-inside-details-of-two-billion-dollar-wme-img-financing.
10 “Independent” in this context makes sense in contrast with “studio movie,” that is, the concentration of the activities of production, distribution, and retail under the unified umbrella of a studio. The practice of “packaging,” traditionally attached to studio production, gets partly redefined and expanded to include the activity of putting together such “big indie films,” with a focus on finding their financing and organizing their distribution. Small independent films made with no money outside of “Big Hollywood” do not belong to this category and do not result from the activity of the same participants.
11 Lynda Obst, Sleepless in Hollywood: Tales from the New Abnormal in the Movie Business (New York: Simon & Schuster, 2013); Roussel and Bielby, eds., Brokerage and Production.
12 In 2013. Motion Picture Association of America, Theatrical Market Statistics , 2013, p. 4, www. mpaa.org/wp-content/uploads/2014/03/MPAA-Theatrical-Market-Statistics-2013_032514-v2.pdf.
13 Midsize agencies practice “copackaging” with larger companies that have more star clients.
14 Indie film agent, big agency, Los Angeles, 2013.
15 “There is a bunch of things we do, not just financing movies. . . . We’ll make investments in different movies that only come after the fact in North America. So it has nothing to do with me packaging a movie, it has to do with me saying, “Okay, I like that as an investment, let me see if one of my guys wants to put money in.” We do all of the financial analysis and put them as producers on those movies” (indie film agent, midsize agency, 2013, his emphasis).
16 Such dematerialization mechanisms also generate more “spec work” for agents, to use the category forged by John Caldwell (see chapter 2).
17 This is also true of the agents who work in the countless small companies of Little Hollywood, although it takes somewhat different forms in their case: directly impacted by the precarization of labor affecting their clients and the subsequent diminution in revenue for their agency, they sometimes have to jump from one small boutique to another at the mercy of economic ups and downs that can quickly lead a small organization to close doors. Their fear of becoming “dispensable” (in one person’s words) or irrelevant also stems from the experience of extensive technological changes: the new electronic tools that equip the practice of agenting increasingly turn it into the rationalized management of massive lists of clients (whose profiles are submitted mechanically to casting professionals via online platforms that disclose the information about available jobs to all agencies at once) and devaluate the “craft of agenting” as the agents knew it (getting the information first based on personal connections within the casting community, having more time to advocate for individual clients, and so on). However, even the agents in the most vulnerable positions claim the ideal of the fearless entrepreneur-agent who thrives in tumultuous times, dissociating themselves from a critique of flexibility or from making common cause with precarized clients. For convergent observations, see Berg and Penley, as well as Mayer, chapters 4 and 11.
18 Lucien Karpik, Valuing the Unique: The Economics of Singularities (Princeton, NJ: Princeton University Press, 2010).


5.3: Respiration
Respiration Overview
In the next few modules we start to learn about the process of respiration and the roles that electron transport chains play in this process. A definition of the word "respiration" that most people are familiar with is "the act of breathing". When
we
breath, air-including molecular oxygen- is brought into our lungs from outside of the body, the oxygen then becomes reduced, and waste products, including the reduced oxygen in the form of water, are exhaled. Simply put, some reactant comes into the organism and then gets reduced and leaves the body as a waste product. This generic idea can be generally applied across biology and oxygen need not always be the compound that brought in, reduced, and dumped as waste. These compounds where the electron is finally "dumped" are more generally known as "
terminal electron acceptors
". The molecules
from which
the electrons originate can vary greatly across biology (we have looked at one common source - the reduced carbon-based molecule glucose). We will often refer to these molecules as "fuel", and the analogy with the burning of gasoline, propane, or coal is quite apt; these are all simply sources of electrons, and the waste product is CO
2
and water. But once again, keep in mind that many organisms can use non-carbon based sources of electrons, and transfer the electrons to other kinds of oxidizing agents.
In between the original electron source and the terminal electron acceptor are a series of biochemical reactions involving at least one redox reaction (during the course of glycolysis, pyruvate oxidation, and the TCA cycle we saw many!). These redox reactions harvest energy for the cell by coupling exergonic redox reactions to an energy-requiring reactions in the cell. In respiration, a special set of enzymes carry out a linked series of redox reactions that ultimately transfer electrons to the terminal electron acceptor. These "chains" of redox enzymes and electron carriers are called electron transport chains (ETC). ETCs are therefore the portion of respiration that use an electron acceptor (brought in from outside of the cell) as the final/terminal acceptor for the electrons that were removed from the intermediate compounds in catabolism. In aerobically respiring eukaryotic cells the ETC is composed of four large, multiprotein complexes embedded in the inner mitochondrial membrane and two small diffusible electron carriers shuttling electrons between them. The electrons are passed from enzyme to enzyme through a series of redox reactions. These reactions couple the exergonic redox transfers to the endergonic transport of hydrogen ions across the membrane. This process contributes to the creation of a transmembrane electrochemical gradient. The electrons passing through the ETC gradually lose potential energy, and are finally deposited on the terminal electron acceptor, which is removed as waste from the cell. When oxygen is the final electron acceptor, the free energy difference of this multistep redox process- from the 1st step in the ETC to the last- is ~ -60 kcal/mol when NADH donates electrons or 45 kcal/mol when FADH
2
donates electrons.
Introduction to redox, oxidative phosphorylation and Electron Transport Chains
In a prior module (Cashing in on Redox) we discussed the general concept of redox reactions in biology and introduced the Electron Tower, a tool to help you understand redox chemistry and to estimate the direction and magnitude of potential energy differences for various redox couples. In earlier modules, substrate level phosphorylation and fermentation were discussed and we saw how exergonic redox reactions could be directly coupled by enzymes to the endergonic synthesis of ATP. These processes are hypothesized to be one of the oldest forms of energy production used by cells. In this section we discuss the next evolutionary advancement in cellular energy metabolism, oxidative phosphorylation of ADP to form ATP. First and foremost, oxidative phosphorylation does not necessarily imply the use of oxygen as the terminal electron acceptor. Oxidative phosphorylation evolved before O 2 was available, and has continued to evolve since then. As "obligate aerobes" (organisms that can't live without O 2 ) we ourselves can only use O 2 as the terminal acceptor. Some organisms are far more flexible, metabolically, and others (obligate anaerobes) can't use O 2 at all. The process of generating ATP from ADP is called oxidative phosphorylation (vs. the substrate level phosphorylation seen earlier, in glycolysis and the citric acid cycle) because it relies on redox reactions to generate a electrochemical transmembrane potential that can then be used by the cell to add a phosphate group to ADP (or to do other kinds of work). If you think about it, these are not the most appropriate names, as redox reactions occur in glycolysis and the citric acid cycle too.
A quick summary of Electron Transport Chains
The ETC begins with the donation of electrons from NADH, FADH 2 or other reduced compounds to a first acceptor in the chain. The electron moves through a series of electron transporters, carriers (mostly proteins) that undergo a series of redox reactions. The free energy transferred from these exergonic redox reactions is, in some steps, coupled to the endergonic movement of protons across a membrane. Since the membrane is an effective barrier to ions, this pumping results in an unequal accumulation of protons across the membrane. This in turn "polarizes" or "charges" the membrane, with a net positive (protons) on one side of the membrane and a negative charge on the other side of the membrane. The separation of charge creates an electrical potential . In addition, the accumulation of protons also causes a pH gradient known as a chemical potential across the membrane (the protons are being pumped from a low concentration to a higher [H + ] environment). Together these two gradients (electrical and chemical) are called an electro-chemical gradient . The -∆G of the electrons moving down the ETC is harvested in the form of this gradient.
Review: The Electron Tower
Since redox chemistry is so central to the topic we begin with a quick review of the table of reduction potentials - sometimes called the "redox tower" or "electron tower". You may hear your instructors use these terms interchangeably. As we discussed in previous modules, all kinds of compounds can participate in biological redox reactions. Making sense of all of this information and ranking potential redox pairs can be confusing. A tool has been developed to rate redox half reactions based on their reduction potentials or E 0 ' values. Whether a particular compound can act as an electron donor (reductant) or electron acceptor (oxidant) depends on what other compound it is interacting with. The redox tower ranks a variety of common compounds (their half reactions) from most negative E 0 ' , compounds that readily get rid of electrons , to the most positive E 0 ' , compounds most likely to accept electrons . The tower organizes these half reactions based on the ability of comopunds to accept electrons. In many redox towers each half reaction is written by convention with the oxidized form on the left followed by the reduced form to its right. The two forms may be either separated by a slash, for example the half reaction for the reduction of NAD + to NADH is written: NAD + /NADH (2e - ), or by separate columns. An electron tower is shown below.
Note
Use the redox tower above as a reference guide to orient you as to the reduction potential of the various compounds. Redox reactions may be either exergonic or endergonic depending on the relative redox potentials of the donor and acceptor (represented by two different half reactions on the tower). Also remember there are many different ways of looking at this conceptually; this type of redox tower is just one way.
Note: Language shortcuts reappear
In the redox table above some entries seem to be written in unconventional ways. For instance Cytochrome c ox/red . There only appears to be one form listed. Why? This is another example of a language shortcut (probably employed so it fits in the column) that can be confusing. The notation above could be rewritten as Cytochrome c ox /Cytochrome c red to indicate that the cytochrome c protein can exist in either an oxidized state (Cytochrome c ox ) or reduced state (Cytochrome c red ).
Review Redox Tower Video
For a short video on how to use the redox tower in redox problems click here . This video was made by Dr. Easlon for Bis2A students.
Using the redox tower: A tool to help understand electron transport chains
By convention the tower half reactions are written with the oxidized form of the compound on the left and the reduced form on the right. Notice that compounds such as glucose and hydrogen gas are excellent electron donors and have very low reduction potentials E 0 ' . Compounds, such as oxygen and nitrite, whose half reactions have relatively high positive reduction potentials (E 0 ' ) generally make good electron acceptors and are found at the opposite end of the table.
Example: Menaquinone
Let's look at menaquinone ox/red . This compound sits in the middle of the redox tower with an half-reaction E 0 ' value of -0.074 eV. Menaquinone ox can spontaneously (ΔG<0) accept electrons from reduced forms of compounds with lower half-reaction E 0 ' . Such transfers form menaquinone red and the oxidized form of the original electron donor. In the table above, examples of compounds that could act as electron donors to menaquinone include FADH 2 , an E 0 ' value of -0.22, or NADH, with an E 0 ' value of -0.32 eV. Remember the reduced forms are on the right hand side of the red/ox pair.
Once menaquinone has been reduced to menaquinone(red), it can now spontaneously (ΔG<0) donate electrons to any compound with a higher half-reaction E 0 ' value. Possible electron acceptors include cytochrome b ox with an E 0 ' value of 0.035 eV; or ubiquinone ox with an E 0 ' of 0.11 eV. Remember that the oxidized forms lie on the left side of the half reaction.
Electron Transport Chains
An electron transport chain , or ETC , is composed of a group of protein complexes in and around a membrane that help energetically couple a series of exergonic/spontaneous redox reactions to the endergonic pumping of protons across the membrane to generate a an electrochemical gradient. This electrochemical gradient creates a free energy potential that is termed a proton motive force whose energetically "downhill" exergonic flow can later be coupled to a variety of cellular processes.
ETC Overview
Step 1. Electrons enter the ETC from an electron donor, such as NADH or FADH 2 which are generated during a variety of catabolic reactions, including those associated with glucose oxidation. Depending on the complexity (number and types of electron carriers) of the ETC being used by an organism, electrons can enter at a variety of places in the electron transport chain - this depends upon the respective reduction potentials of the proposed electron donors and acceptors. In some metabolically flexible organisms, the chain could also terminate at a variety of external electron acceptors.
Step 2. After the first redox reaction, the initial electron donor will become oxidized and the electron acceptor will become reduced. The difference in redox potential between the electron acceptor an donor is related to ΔG by the relationship ΔG = -nFΔE, where n = the number of electrons transferred and F = Faraday's constant. The larger a positive ΔE the more exergonic a reaction.
Step 3. If sufficient energy transferred during an exergonic redox step the electron carrier may couple this negative change in free energy to the endergonic process of transporting a proton from one side of the membrane to the other.
Step 4. After multiple redox transfers, the electron is delivered to a molecule known as the terminal electron acceptor. In the case of humans, the terminal electron acceptor is oxygen. However, there are many, many, many, other possible electron acceptors, see below.
Note
Electrons entering the ETC do not have to come from NADH or FADH 2 . Many other compounds can serve as electron donors, the only requirements are that there exists an enzyme that can oxidize the electron donor and then reduce another compound and that the E 0 ' be positive (e.g. ΔG<0). Even a small amounts of free energy transfers can add up. For example there are bacteria that use H 2 as an electron donor. This is not too difficult to believe because the half reaction 2H + + 2 e - /H 2 has a reduction potential (E 0 ') of -0.42 V. If these electrons are eventually delivered to oxygen then the ΔE 0 ' of the reaction is 1.24 V which corresponds to a large negative ΔG (-ΔG). Alternatively, there are some bacteria that can oxidize iron, Fe 2 + at pH 7 to Fe 3 + with a reduction potential (E 0 ') of + 0.2 V. These bacteria use oxygen as their terminal electron acceptor and in this case, the ΔE 0 ' of the reaction is approximately 0.62 V. This still produces a -ΔG. The bottom line is that depending on the electron donor and acceptor that the organism uses, a little or a lot of energy (per electron) can be transferred and used by the cell as electrons are donated to the electron transport chain.
How do you calculate ∆E? It's pretty simple using the redox table- look for the final state (in V) of the electron (say, water, from reduction of O2) and subtract the initial state (say, as glucose) . Using this chart your calculation ∆E would be 0.82 -(-0.43) = 1.25 volts. You do not need to worry (yet) about the number of electrons as ∆ volts have units of ∆ unit energy/unit charge (for example, Joules per Coulomb).
What are the complexes of the ETC?
ETCs are made up of a series (or at least one) of membrane associated redox proteins or (some are integral) protein complexes (complex = more than one protein arranged in a quaternary structure) that move electrons from a donor source, such as NADH, to a final terminal electron acceptor, such as oxygen. This particular donor/terminal acceptor pair is the primary one used in human mitochondria. Each electron transfer in the ETC requires a reduced substrate as an electron donor and an oxidized substrate as the electron acceptor. In most cases the electron acceptor is a member of the enzyme complex. Once the complex is reduced, the complex can serve as an electron donor for the next reaction.
How do ETC complexes transfer electrons?
As previously mentioned the ETC is composed of a series of protein complexes that undergo a series of linked redox reactions. These complexes are in fact multiprotein enzyme complexes referred to as oxidoreductases: the enzymes perform both oxidation (of the step in the chain "above" them, energetically) and reduction (of the complex in the complex in the chain below them) or reductases (again, in spite of the fact that the enzyme both oxidizes an electron donor and passes that electron to an electron acceptor). The one exception to this naming convention is the terminal complex in aerobic respiration that uses molecular oxygen as the terminal electron acceptor. That enzyme complex is referred to as an oxidase (even though it both oxidizes and reduces something). Redox reactions within these complexes are typically carried out by a non-protein component called a prosthetic group . This is true for all of the electron carriers with the exception of quinones, which are a class of lipids that can directly be reduced or oxidized by the oxidoreductases. In this case, both the Quinone red and the Quinone ox is soluble within the membrane and can move from complex to complex. The prosthetic groups are directly involved in the redox reactions being catalyzed by their associated oxidoreductases. In general these prosthetic groups can be divided into two general types: those that carry both electrons and protons and those that only carry electrons.
Question:
If the redox reaction is carried out by a non-proteinaecious "prosthetic group" (see list below)- why bother with the protein part of the complex?
The Electron and Proton carriers in the ETC
- Flavoproteins (Fp) , these proteins contain an organic prosthetic group called a flavin , which is the actual moiety that undergoes the oxidation/reduction reaction. FADH 2 is an example of a Fp.
- Quinones , are a family of lipids which means they are soluble within the membrane.
- NADH and NADPH are electron (2e-) and proton (2 H + ) carriers, of course. The FADH 2 we know from the citric acid cycle is actually bound to the enzyme succinate dehydrogenase.
- Cytochromes are proteins that contain a heme prosthetic group. The Heme is capable of carrying a single electron.
- Iron-Sulfur proteins contain a non-heme iron-sulfur clusters that can carry an electron. The prosthetic group is often abbreviated as Fe-S
Aerobic versus Anaerobic respiration
In the world we live in, most of the organisms we see around us breath air, which is approximately 20% oxygen. Oxygen is our terminal electron acceptor . We call this process respiration, specifically aerobic respiration. We breath in oxygen, it diffuses into our cells and to our mitochondria where it is used as the final acceptor of electrons from our electron transport chains. That is aerobic respiration : the process of using oxygen as a terminal electron acceptor in an electron transport chain.
While we may use oxygen as the terminal electron acceptor for our respiratory chains, the more general process of respiration evolved at time when oxygen was not a major component of the atmosphere. Respiration or oxidative phosphorylation does not require oxygen at all; it simply requires a compound with a high reduction potential to act as a terminal electron acceptor. Many organisms can use a variety of compounds including nitrate (NO 3 - ), nitrite (NO 2 - ), even iron (Fe +++ ) as terminal electron acceptors. When oxygen is NOT the terminal electron acceptor, the process is referred to as anaerobic respiration . The ability of organisms to vary their terminal electron acceptor provides metabolic flexibility and can ensure better survival if any given terminal acceptor is in limited supply. Think about this: in the absence of oxygen we die; but an organism that can use a different terminal electron acceptor can survive.
Question: What's the difference between anaerobic respiration and fermentation?
Here is a fun video in which a nice young man COMPLETELY FAILS TO UNDERSTAND what anaerobic respiration is, teaching us instead about fermentation. Beware of the internet! I'd say "you get what you pay for" but this is actually a site you have to pay for!
A generic example of a simple, 2 complex ETC
The figure below depicts a generic electron transport chain, composed of two integral membrane complexes; Complex I ox and Complex II ox . A reduced electron donor, designated DH (such as NADH or FADH 2 ) reduces Complex I ox giving rise to the oxidized form D (such as NAD + or FAD). Simultaneously, a prosthetic group within complex I is now reduced (accepts the electrons). In this example the redox reaction is exergonic and the free energy difference is coupled by the enzymes in Complex I to the endergonic translocation of a proton from one side of the membrane to the other. The net result is that one surface of the membrane becomes more negatively charged, due to an excess of hydroxyl ions (OH - ) and the other side becomes positively charged due to an increase in protons on the other side. Complex I red can now reduce the prosthetic group in Complex II red while simultaneously becoming oxidized by Complex II ox . Electrons pass from Complex I to Complex II via thermodynamically spontaneous redox reactions, regenerating Complex I ox which can repeat the previous process. Complex II red reduces A, the terminal electron acceptor to regenerate Complex II ox and create the reduced form of the terminal electron acceptor. In this case, Complex II can also translocate a proton during the process. If A is molecular oxygen, water (AH) will be produced. When A is oxygen the reaction scheme would be considered a model of an aerobic ETC. However, if A is nitrate, NO 3 - then Nitrite, NO 2 - is produced (AH) and this would be an example of an an aerobic ETC.
Generic 2 complex electron transport chain. In the figure, DH is the electron donor (donor reduced) and D is the donor oxidized. A is the oxidized terminal electron acceptor and AH is the final product, the reduced form of the acceptor. As DH is oxidized to D, protons are translocated across the membrane, leaving an excess of hydroxyl ions (negatively charged) on one side of the membrane and protons (positively charged) on the other side of the membrane. The same reaction occurs in Complex II as the terminal electron acceptor is reduced to AH.
Exercise 1
Thought question
Based on the figure above and using an electron tower, what is the difference in the electrical potential if (A) DH is NADH and A is O 2 and (B) DH is NADH and A is NO 3 - . Which pairs (A or B) provide the most amount of usable energy?
In the nitrate vs nitrite example above, both the oxidized and the reduced forms have a charge of minus 1. So... how to you know which form is more oxidized?
The figure above has a problem- it depicts an electron floating within the membrane. This wouldn't happen- the membrane is not a "wire", it is an "insulator". You might draw your own corrected version.
Detailed look at aerobic respiration
The eukaryotic mitochondria has evolved a very efficient ETC. There are four complexes composed of multiple proteins and their prosthetic groups. These complexes labeled I through IV depicted in the figure below. The aggregation of these four complexes, together with associated mobile, accessory electron carriers, makes up the electron transport chain. This type of electron transport chain is present in multiple copies in the inner mitochondrial membrane of eukaryotes.
The electron transport chain is a series of electron transporters embedded in the inner mitochondrial membrane that shuttles electrons from NADH and FADH 2 to molecular oxygen. In the process, protons are pumped from the mitochondrial matrix to the intermembrane space, and oxygen is reduced to form water.
Complex I
To start, two electrons are carried to the first complex aboard NADH. This complex, labeled I, is composed of flavin mononucleotide (FMN) and an iron-sulfur (Fe-S)-containing protein. FMN, which is derived from vitamin B 2 , also called riboflavin, is one of several prosthetic groups or co-factors in the electron transport chain. These prosthetic groups (non-protein molecule required for the activity of a protein) are organic or inorganic non-peptide molecules bound to a protein, covalently or noncovalently; prosthetic groups include co-enzymes, which are the prosthetic groups of enzymes. The enzyme in complex I is NADH dehydrogenase (aka NADH Q reductase) and is a very large protein, containing 45 amino acid chains. Complex I can pump four hydrogen ions across the membrane from the matrix into the intermembrane space, and it is in this way that the hydrogen ion gradient is established and maintained between two aqueous compartments separated by the inner mitochondrial membrane.
Q and Complex II
Complex II directly receives FADH 2 , FADH 2 's electrons do not pass through complex I. The compound connecting the first and second complexes to the third is ubiquinone (Q). The Q molecule is lipid-soluble and freely moves through the hydrophobic core of the membrane. Once it is reduced, (QH 2 ), ubiquinone delivers its electrons to the next complex in the electron transport chain. Q receives the electrons derived from NADH from complex I and the electrons derived from FADH 2 from complex II, including succinate dehydrogenase. This enzyme and FADH 2 form a small complex that delivers electrons directly to the electron transport chain, bypassing the first complex. Since these electrons bypass and thus do not energize the proton pump in the first complex, fewer ATP molecules are made from the FADH 2 electrons. As we will see in the following section, the number of ATP molecules ultimately obtained is proportional to the number of protons pumped across the inner mitochondrial membrane.
Complex III
There's a lot of redox going on in this picture. Which is the upstream donor of electrons coming into Complex III? Which complex/molecule does Complex III reduce?
The third complex is composed of proteins that carry a number of prosthetic groups: a B-type cytochrome, a c-type cytochrome, and an Iron-sulfur cluster; this complex is called cytochrome bc 1 complex. Cytochrome (= "cellular pigment") proteins have a prosthetic group that includes a metal bound by heme. The heme molecule is similar to the heme in hemoglobin, but it carries electrons, not oxygen. As a result, the iron ion at its core is reduced and oxidized as it passes the electrons, fluctuating between different oxidation states: Fe ++ (reduced) and Fe +++ (oxidized). The heme molecules in the cytochromes have slightly different characteristics due to the effects of the different proteins binding them, giving slightly different characteristics to each complex. Complex III pumps protons through the membrane and passes its electrons to cytochrome c for transport to the fourth complex of proteins and enzymes (cytochrome c is the acceptor of electrons from Q; however, whereas Q carries pairs of electrons, cytochrome c can accept only one at a time).
Cytochrome C
The cytochrome complex , or cyt c is a small hemeprotein found loosely associated with the inner membrane of the mitochondrion. It belongs to the cytochrome c family of proteins. Cytochrome c is an essential component of the electron transport chain, where it carries one electron. It is capable of undergoing oxidation and reduction as its iron atom converts between the ferrous and ferric forms, but does not bind oxygen. It transfers electrons between Complexes III (Coenzyme Q – Cyt C reductase) and IV (Cyt C oxidase).
Complex IV, cytochrome C oxidase
This complex contains two heme groups (one in each of the two cytochromes, a, and a3) and three copper ions (a pair of CuA and one CuB in cytochrome a3). The cytochromes hold an oxygen molecule very tightly between the iron and copper ions until the oxygen is completely reduced (the donor, cytochrome C, can only deliver one electron at a time). The reduced oxygen then picks up two hydrogen ions from the surrounding medium to make water (H 2 O). The removal of the hydrogen ions from the system contributes to the ion gradient used in the process of oxidative phosphorylation (as well as providing the -∆G required to drive other endergonic processes).
Here is a very nice video that includes the electron transport chain- I think it's very helpful, check it out! It starts with ATP synthase, which we haven't discussed yet, but that's fine, it'll explain why the cell is building that proton gradient!
I have one problem with this animation:There's really no discussion of how proton pumping works- the discussion's extremely vague- one might even come away with the notion that a gas forms within the matrix-domain of NADH dehydrogenase (complex I). In fact, the proton pump of complex I is entirely embedded within the membrane and isn't illustrated here at all. Presumably this is because the mechanism of pumping is a subject of debate. For the purposes of this class, you simply need to know- as mentioned in the video- that the endergonic formation of a proton gradient is powered by the exergonic redox reactions. In NADH dehyrogenase, this includes the oxidation of NADH by a flavin within the enzyme, some internal transfers of electrons within the enzyme, and then the transfer of the electron to a membrane-soluble quinone (coenzyme Q), to form a quinol. Each of these 2-electron transfers is illustrated below, with the oxidized molecules shown first.
Question:
Could NADH dehyrodrogenase reduce NAD + as well as oxidize NADH? Under what circumstances might this happen?
Chemiosmosis
The free energy from the series of redox reactions just described is used to pump hydrogen ions (protons) across the membrane, resulting in a concentration of protons on one side of the membrane. The uneven distribution of H + ions across the membrane establishes both concentration and electrical gradients (thus, an "electrochemical" gradient), owing to the hydrogen ions’ positive charge and their aggregation on one side of the membrane.
If the membrane were open to diffusion by the hydrogen ions, the ions would tend to diffuse back across into the matrix, driven by their electrochemical gradient. Many ions cannot diffuse through the nonpolar regions of phospholipid membranes without the aid of ion channels. Similarly, hydrogen ions in the matrix space can only pass through the inner mitochondrial membrane via an integral membrane protein called ATP synthase (depicted below, and illustrated in the video recommended above). This complex acts as a tiny generator, turned by the protons moving down their electrochemical gradient. The movement of this molecular machine serves to both lower the activation energy of the ADP + Pi -> ATP reaction and couple the exergonic transfer of energy associated with the movement of protons down their electrochemical gradient to the endergonic addition of a phosphate to ADP, forming ATP.
ATP synthase is a complex, molecular machine that uses a proton (H + ) gradient to form ATP from ADP and inorganic phosphate (Pi). (Credit: modification of work by Klaus Hoffmeier). If you have had some O-chem and would like to dig a little deeper into the mechanism through which the mechanical flexing of this complex promotes ATP formation, check out this video .
Suggested Discussion
Dinitrophenol (DNP) is a small chemical that makes the membrane leaky to protons. It is sometimes used as a weight-loss drug, and can have lethal consequences . What effect would you expect DNP to have on the difference in pH across both sides of the inner mitochondrial membrane? Why do you think this might be an effective weight-loss drug? Why might it be dangerous?
In our cells, chemiosmosis is used to generate 90 percent of the ATP made during aerobic glucose catabolism; it is also the method used in the light reactions of photosynthesis to harness the energy of sunlight in the process of photophosphorylation (vs. oxidative phosphorylation via the ATPase of mitochondria). The overall result of these reactions is the production of ATP from the energy of the electrons removed from hydrogen atoms. These atoms were originally part of a glucose molecule. At the end of the pathway, the electrons are used to reduce an oxygen molecule to water.
Super-helpful video:
How ATP is made by ATP synthase
In oxidative phosphorylation, the pH gradient formed by the electron transport chain is used by ATP synthase to form ATP in a Gram- bacteria.
Suggested Discussion
Cyanide inhibits cytochrome C oxidase, a component of the electron transport chain. If cyanide poisoning occurs, would you expect the concentration of protons of the intermembrane space to increase or decrease? What effect would cyanide have on ATP synthesis?


25.2: Approximated Hamiltonians
In order to solve the TISEq for a many-electron atom we also need to approximate the Hamiltonian, since analytic solution using the full Hamiltonian as in Equation 26.1 are impossible to find. The most significant approximation used in chemistry is called the variational method .
Variational method
The basic idea of the variational method is to guess a “trial” wave function for the problem consisting of some adjustable parameters called “variational parameters”. These parameters are adjusted until the energy of the trial wave function is minimized. The resulting trial wave function and its corresponding energy are variational method approximations to the exact wave function and energy.
Why would it make sense that the best approximate trial wave function is the one with the lowest energy? This results from the Variational Theorem, which states that the energy of any trial wave function \(E\) is always an upper bound to the exact ground state energy \({\cal E}_0\). This can be proven easily. Let the trial wave function be denoted \(\Phi\). Any trial function can formally be expanded as a linear combination of the exact eigenfunctions \(\Psi_i\). Of course, in practice, we don’t know the \(\Psi_i\), since we are applying the variational method to a problem we can’t solve analytically. Nevertheless, that doesn’t prevent us from using the exact eigenfunctions in our proof, since they certainly exist and form a complete set, even if we don’t happen to know them. So, the trial wave function can be written:
\[ \Phi = \sum_i c_i \Psi_i, \label{26.2.1} \]
and the approximate energy corresponding to this wave function is:
\[ E[\Phi] = \dfrac{\int \Phi^* {\hat H} \Phi d\mathbf{\tau}}{\int \Phi^* \Phi d\mathbf{\tau}}, \label{26.2.2} \]
where \(\mathbf{\tau}=\left(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N\right)\) is the ensemble of the spatial coordinates of each electron and the integral symbol is assumed as a \(3N\)-dimensional integration. Replacing the expansion over the exact wave functions, we obtain:
\[ E[\Phi] = \dfrac{\sum_{ij} c_i^* c_j \int \Psi_i^* {\hat H} \Psi_jd\mathbf{\tau}}{ \sum_{ij} c_i^* c_j \int \Psi_i^* \Psi_jd\mathbf{\tau}}. \label{26.2.3} \]
Since the functions \(\Psi_j\) are the exact eigenfunctions of \({\hat H}\), we can use \({\hat H} \Psi_j = {\cal E}_j \Psi_j\) to obtain:
\[ E[\Phi] = \dfrac{\sum_{ij} c_i^* c_j {\cal E}_j \int \Psi_i^* \Psi_j d\mathbf{\tau}}{ \sum_{ij} c_i^* c_j \int \Psi_i^* \Psi_j d\mathbf{\tau}}. \label{26.2.4} \]
Now using the fact that eigenfunctions of a Hermitian operator form an orthonormal set (or can be made to do so), we can write:
\[ E[\Phi] = \dfrac{\sum_{i} c_i^* c_i {\cal E}_i}{\sum_{i} c_i^* c_i}. \label{26.2.5} \]
We now subtract the exact ground state energy \({\cal E}_0\) from both sides to obtain
\[ E[\Phi] - {\cal E}_0 = \dfrac{\sum_i c_i^* c_i ( {\cal E}_i - {\cal E}_0)}{ \sum_i c_i^* c_i}. \label{26.2.6} \]
Since every term on the right-hand side is greater than or equal to zero, the left-hand side must also be greater than or equal to zero:
\[ E[\Phi] \geq {\cal E}_0. \label{26.2.7} \]
In other words, the energy of any approximate wave function is always greater than or equal to the exact ground state energy \({\cal E}_0\).
This explains the strategy of the variational method: since the energy of any approximate trial function is always above the true energy, then any variations in the trial function which lower its energy are necessarily making the approximate energy closer to the exact answer. (The trial wave function is also a better approximation to the true ground state wave function as the energy is lowered, although not necessarily in every possible sense unless the limit \(\Phi = \Psi_0\) is reached).
Approximated solution for the helium atom
We now have all the ingredients to attempt the simplest approximated solution to the TISEq of a many-electron atom. We can start by writing the total wave function using the Slater determinant of Equation 26.1.7 in terms of spin-orbitals:
\[ \Psi (\mathbf{x}_{1},\mathbf{x}_{2},\ldots ,\mathbf{x}_{N})= |\chi_{1},\chi_{2},\cdots ,\chi_{N}\rangle = |\psi_{1}\phi_{\uparrow},\psi_{1}\phi_{\downarrow},\cdots ,\psi_{\dfrac{N}{2}}\phi_{\uparrow},\psi_{\dfrac{N}{2}}\phi_{\downarrow}\rangle, \label{26.2.8} \]
and then we can replace it into the TISEq for an \(N\)-electron system. This results into a set of \(N\) one-electron equations, one for each electron. When we attempt to solve each individual equation, however, we end up with a problem, since the potential energy in the Hamiltonian of Equation 26.1 does not have spherical symmetry because of the electron-electron repulsion term. As such, the one-electron TISEq cannot be simply solved in spherical polar coordinates, as we did for the hydrogen atom in chapter 21 . The simplest way of circumventing the problem is to neglect the electron-electron repulsion term (i.e., assume that the electrons are not correlated and do not interact with each other). For a 2-electron atom this procedure is straightforward, since the Hamiltonian can be written as a sum of one-electron Hamiltonians:
\[ \hat{H} =\hat{H}_1+\hat{H}_2, \label{26.2.9} \]
with \(\hat{H}_1\) and \(\hat{H}_2\) looking identical to those used in the TISEq of the hydrogen atom. This one-particle Hamiltonian does not depend on the spin of the electron, and therefore, we can neglect the spin component of the Slater determinant and write the total wave function for the ground state of helium, Equation 26.1.4, simply as:
\[ \Psi({\bf r}_1, {\bf r}_2) = \psi_{100}({\bf r}_1)\psi_{100}({\bf r}_2). \label{26.2.10} \]
The overall TISEq reduces to a set of two single-particle equations:
\[ \begin{aligned} \hat{H}_1 \psi_{100}({\bf r}_1) &= E_1\psi_{100}({\bf r}_1) \\ \hat{H}_2 \psi_{100}({\bf r}_2) &= E_2\psi_{100}({\bf r}_2), \end{aligned} \label{26.2.11} \]
which can then be solved similarly to those for the hydrogen atom, and the solution be combined to give:
\[ E = E_1+E_2. \label{26.2.12} \]
In other words, the resulting energy eigenvalue for the ground state of the helium atom in this approximation is equal to twice the energy of a \(\psi_{100}\), \(1s\), orbital. The resulting approximated value for the energy of the helium atom is \(7,217 \text{ kJ/mol}\), compared with the exact value of \(7,620 \text{ kJ/mol}\).
The nuclear charge \(Z\) in the \(\psi_{100}\) orbital can be used as a variational parameter in the variational method to obtain a more accurate value of the energy. This method provides a result for the ground-state energy of the helium atom of \(7,478 \text{ kJ/mol}\) (only \(142 \text{ kJ/mol}\) lower than the exact value), with the nuclear charge parameter minimized at \(Z_{\text{min}}=1.6875\). This new value of the nuclear charge can be interpreted as the effective nuclear charge that is felt by one electron when a second electron is present in the atom. This value is lower than the real nuclear charge (\(Z=2\)) because the interaction between the electron and the nuclei is shielded by presence of the second electron.
This procedure can be extended to atoms with more than two electrons, resulting in the so-called Hartree-Fock method. The procedure, however, is not straightforward. We will explain it in more details in the next chapter, since it is the simplest approximation that also describes the chemical bond.


CRA is grounded in the constructivist theory of learning. Teachers begin with concrete manipulatives, transition to visual representations/drawings, and then move to abstract mathematical numbers and symbols. When teaching students addition and subtraction, begin with models/manipulatives, and then progress to bar diagrams, and number lines.
Addition and Subtraction Computation
In kindergarten, students developed an understanding of addition as putting together and adding to, and subtraction as taking apart and taking from. Students in first grade are expected to add two-digit numbers to one-digit numbers; and in second grade, students add two- and three-digit numbers. It is important to note that the traditional algorithms are not required until fourth grade. Therefore, students are expected to compose and decompose numbers and use flexible methods to solve problems.
When adding two or more numbers, the result is the
sum
. The
symbol used for addition
is the plus sign (+). The inverse of addition is
subtraction
, and the result is called the
difference
. The symbol used for subtraction is the minus sign (-).
First grade students will extend their number facts and place value strategies to add within 100. Do not introduce the traditional addition or subtraction algorithms in first grade. Instead, represent a problem using words, numbers, pictures, manipulatives, and/or symbols.
Addition and Subtraction with Bar Diagrams
Another model to use when adding and subtraction is the bar diagram. Bar diagrams (sometimes called strip or tape diagrams) are a pictorial representation of a problem shown by bars or boxes. Using bar diagrams is the representational/semi-concrete link between the concrete and the abstract phases of learning..
Note the following examples:
Susan made 24 cookies for the school carnival. Terry made 36 cookies. How many cookies will the school carnival have to share?
Kathryn has read 17 books. Her goal is to read 52 this year. How many more books does she need to read?
Addition and Subtraction with Open Number Lines
An open number line has no numbers or marks on it; it is used when students want to share their thinking strategies. Before using the open number line, students must have a strong understanding of numbers up to 100. Prerequisite skills include counting on and counting back, knowing basic addition and subtraction facts to ten, and adding/subtracting a multiple of 10 to or from any two-digit number.
To introduce the open number line, ask students questions such as:
-
How can you go from 0 to 37 in the least number of jumps of tens and ones?
Encourage students to share their strategies and discuss which strategy is most efficient. For example, when jumping from 0 to 37, one student could say, “three jumps of ten and seven jumps of one” and another student could say, “four jumps of ten and back three ones.”
When students have mastered questions like the one above, the next phase in the use of the open number line is with addition and subtraction. Be sure to give students the flexibility to choose what types of jumps they want to use. Some possible questions are:
-
How can you go from 26 to 52 in a small number of jumps? Who has another strategy?
-
How can you go from 61 to 37 in a small number of jumps? Who has another strategy?
-
How can you solve 27 + 85?
-
How can you solve 68 – 31?
To solve the problem 27 + 68, one student may begin at 27, add 60 and then add 8.
Another student may solve 27 + 68 by adding 3, then add 60, then add 5.
Multiplication and Division Computation
Second grade students will “use addition to find the total number of objects arranged in rectangular arrays” (KSDE, 2017). Rectangular arrays are an arrangement of objects in horizontal rows and vertical columns and provides a visual model for multiplication. One way to introduce students to arrays is to ask them to build all possible arrays with no more than 25 objects, and up to 5 rows and 5 columns (5 x 5 array). In addition, ask students to draw their arrays on grid paper and write two different equations under their array. An example of this is a 5 x 4 array:
-
The equation by rows is 20 = 5 + 5 + 5 + 5
-
The equation by columns is 20 = 4 + 4 + 4 + 4 + 4
Source: 2017 Kansas Mathematics Standards Flip Book 2nd Grade
Students must explore the concept of rectangular arrays using concrete objects, as well as pictorial representations on grid paper. In addition, students will use the
commutative property of addition
since they can add either the rows or columns and get the same answer.
In third grade, students will find the products of whole numbers and recognize multiplication as finding the total number of objects when there are an equal number of groups. Refer to “groups of” objects, such as 4 x 7 is 4 groups of 7. Begin using the vocabulary terms
factor
and
product
at this level.
Be sure to refer to Table 2 in the K
ansas Mathematics Standards
for the types of multiplication problems.
Students must experience all three types of multiplication problems: size of group is unknown, unknown product, and number of groups is unknown. When students are solving problems, ask them to also write the related multiplication or division equation. For example, to determine the unknown number in the equation 24 ? = 3, students should use their knowledge of the related multiplication fact 3 x 8 = 24.
Watch this video from Graham Fletcher,
“The Progression of Multiplication”
When teaching division, focus on two models of division: partitive and measurement.
-
Partitive division
focuses on the question, “How many are in each group?” An example of the partition model in context is, “There are 15 cookies on the counter. If you are sharing the cookies equally among 5 friends, how many cookies will each friend receive?”
-
Measurement division
(sometimes called repeated subtraction) focuses on the question, “How many groups can you make?” An example of the measurement model is, “There are 15 cookies on the counter. If I give each person 3 cookies, how many people can I give cookies to?”
In third grade, students should be introduced to the vocabulary terms for division:
quotient
,
dividend
, and
divisor
. In addition, students should use models and concrete objects to justify their thinking.
In third grade, students use various strategies to solve word problems. Expect students to use a variety of representations when solving problems, such as rectangular arrays, drawing pictures of equal groups, mental math, number lines, and equations.
Watch this video from Graham Fletcher,
“The Progression of Division.”
Examples of Multiple Strategies for Teaching Multiplication in Third Grade:
There are 24 books to be displayed on the library shelves. If the books are displayed with 4 books on each shelf, how many shelves would be needed?
Rectangular Array
Mental Mathematics
A student could reason that “6 and 6 are 12” and “12 and 12 are 24.” Therefore, there are 4 groups of 6 for a total of 24 books.
Number Line
A number line can show the jumps of equal distance.
Equations
Students may use related equations, such as
-
4 x 6 = ?
-
6 x 4 = ?
-
24 × ? = 4
-
24 ÷ 6 = ?
-
? ÷ 4 = 24
-
? ÷ 6 = 4
Furthermore, third graders should be introduced to
variables
. Letters for variables should begin being used in third grade for the unknown number, for example, 24 ÷
x
= 4.
When students use various strategies and multiple representations, you are helping them develop
multiplicative reasoning
. Multiplicative reasoning is the ability to work flexibly with the concepts, strategies and representations of multiplication (and division). It helps students see the different kinds of relationships between numbers, whereas additive thinking only requires a capacity to work with the numbers themselves (Siemon, 2015). Multiplicative reasoning is related to iterating, or making multiple copies.
Examples of Multiple Strategies for Teaching Division in Third Grade:
The two strategies students can use to solve division word problems are using diagrams and/or pictures (partitive division) or determining the number of shares (measurement division).
This is an example of partitive division since the size of the groups is unknown.
Shelby has 36 candies to share equally with her 4 friends. How many candies will each person receive?
This is an example of measurement division since the number of groups is unknown.
Amie loves sodas and buys one every afternoon. Each soda costs $2. If she has $12, how many days can she buy a soda?
|
Starting
|
Day 1
|
Day 2
|
Day 3
|
Day 4
|
Day 5
|
Day 6
|
|
$12
|
12 – 2 = 10
|
10 – 2 = 8
|
8 – 2 = 6
|
6 – 2 = 4
|
4 – 2 =2
|
2 – 2 = 0
|
Therefore, it took 6 days for Amie to spend her $12 on sodas.
Properties of Multiplication
Students in third grade are expected to understand the properties of multiplication, although they do not need to use the formal terms. Teachers must use the correct terms.
The four mathematical properties of multiplication that students need to be familiar with and use:
-
Commutative Property of Multiplication
,
-
Associative Property of Multiplication
,
-
Multiplicative Identity Property
, and
-
Distributive Property of Multiplication over Addition
.
The
Commutative Property of Multiplication
is when two numbers are multiplied, the product is the same regardless of the order of the factors. To “commute” means to travel, and according to the Commutative Property of Multiplication, the numbers can “move around” without changing the product. For example, the array 3 x 4 can also be shown as the array 4 x 3.
The
Associative Property of Multiplication
is when three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. “To associate” means to connect or join with something; and according to the Associative Property of Multiplication, numbers can “associate” with any other number. For example, 6 x 5 x 2 could be solved the following ways:
(6 x 5) x 2
-
Multiply 6 x 5 first = 30
OR 6 x (5 x 2)
-
Multiply 5 x 2 first = 10
-
Then 30 x 2 = 60 Then 6 x 10 is 60.
The
Distributive Property of Multiplication over Addition
is when multiplying a sum by a number is the same as multiplying each addend separately and then adding the products. To “distribute” means to spread out equally. For example, 8 x 7 could be shown in a rectangular array and then decompose 7 into 5 and 2.
Source: Kansas Mathematics Standards Flip Book 3rd grade (2017a)
Fact Triangles
Fact Triangles can be used to help students see the inverse relationships between multiplication and division.
2 x 7 = 14 7 x 2 = 14
14 ÷ 2 = 7 14 ÷ 7 = 2
2 = 14 ÷ 7 7 = 14 ÷ 2
Click here for
Fact Triangles printables
.]
By the end of third grade, students will
fluently
multiply and divide single digit numbers within 100. This does not mean a focus on timed tests, but focus on hands-on experiences, working with manipulatives and drawing pictures. In addition, give students multiple opportunities to experience multiplication and division problems written both horizontally and vertically. Refer to chapter 5 for strategies students can use to develop fluency with basic multiplication and division facts.
Estimation
Don’t teach estimation by asking students “make a guess.” The word “estimate” comes from the Latin word aestimare, which means to value. To estimate means to judge the value of something, and an estimate is the resulting calculation. A “guess” is to form an opinion with little or no evidence. When you tell students to make a guess, they are making uninformed conclusions. On the other hand, estimating means to make an informed decision based on data and models (Nichol, n.d.).
Teaching Traditional Algorithms
Traditional algorithms are digit oriented whereas invented strategies are number oriented. When students invent strategies for solving problems, they are more flexible with the problems. With the traditional algorithm, students see and do the problem following rules. Additionally, with invented strategies, students make fewer errors.
The Kansas Mathematics Standards suggest that students have knowledge of standard/traditional algorithms in fourth grade for addition and subtraction, fifth grade for multiplication, and sixth grade for division. These topics are introduced much earlier than the grade level for the standard algorithm so that students can make sense of why that algorithm works. The key is that students understand the mathematical operations before they learn the “rules” of traditional algorithms.


13: Microsoft Access Intermediate Skills Last updated Save as PDF Page ID 46389 Learning Objectives Discuss table relationships Work with tables Work with datasheets Sort and filter data 13.1: Why It Matters- Microsoft Access Intermediate Skills 13.2: Introduction to Table Relationships 13.3: Database Principles 13.4: Flat File vs Normalized Data 13.5: Types of Table Relationships 13.6: Introduction to Working with Tables 13.7: Datasheets 13.8: Entering Data 13.9: Changing Values 13.10: Introduction to Working with Datasheets 13.11: Adding Values and Records in Datasheets 13.12: Deleting Records in Datasheets 13.13: Introduction to Sorting and Filtering Data 13.14: Displaying Records in Datasheets 13.15: Sorting Data 13.16: Aggregating Data 13.17: Printing Records 13.18: Putting It Together- Microsoft Access Intermediate Skills 13.19: Assignment- Modify Access Database 13.20: Discussion- Moving Columns and Fields